Find Inverse of Square Root Functions

This tutorial provides step-by-step examples on how to find the inverse of square root functions, including how to determine their domain and range.

Example 1

Find the inverse function, its domain, and range of the function:

\( f(x) = \sqrt{x - 1} \)

Solution:

Domain of \(f\): \([1, +\infty)\), Range of \(f\): \([0, +\infty)\).
Write \(f\) as an equation: \( y = \sqrt{x - 1} \)
Square both sides: \( y^2 = x - 1 \)
Solve for \(x\): \( x = y^2 + 1 \)
Swap \(x\) and \(y\) to get the inverse: \( f^{-1}(x) = x^2 + 1 \)
Domain and range of \(f^{-1}\): Domain: \([0, +\infty)\), Range: \([1, +\infty)\)

Find the inverse, its domain, and range of:

\( f(x) = \sqrt{x + 3} - 5 \)

Solution:

Domain: \(x + 3 \ge 0 \Rightarrow x \ge -3\) → \([-3, +\infty)\)
Range: \([-5, +\infty)\)
Write as equation: \( y = \sqrt{x + 3} - 5 \Rightarrow \sqrt{x + 3} = y + 5 \)
Square both sides: \( x + 3 = (y + 5)^2 \)
Solve for \(x\): \( x = (y + 5)^2 - 3 \)
Swap \(x\) and \(y\): \( f^{-1}(x) = (x + 5)^2 - 3 \)
Domain and range of \(f^{-1}\): Domain: \([-5, +\infty)\), Range: \([-3, +\infty)\)

Find the inverse, its domain, and range of:

\( f(x) = -\sqrt{x^2 - 1}, \quad x \le -1 \)

Solution:

Domain: \((-\infty, -1]\), Range: \((-\infty, 0]\)
Write as equation: \( y = -\sqrt{x^2 - 1} \)
Square both sides: \( y^2 = x^2 - 1 \Rightarrow x^2 = y^2 + 1 \)
Select solution using domain: \( x = -\sqrt{y^2 + 1} \)
Swap \(x\) and \(y\) for the inverse: \( f^{-1}(x) = -\sqrt{x^2 + 1} \)
Domain and range of \(f^{-1}\): Domain: \([0, +\infty)\), Range: \((-\infty, -1]\)

Find the inverse, its domain, and range of:

Answers:

\( f^{-1}(x) = \frac{1}{4}(x + 6)^2 - 2 \), Domain: \((-\infty, -6]\), Range: \([-2, +\infty)\)
\( g^{-1}(x) = \sqrt{\frac{(x - 4)^2}{4} + 4} \), Domain: \([4, +\infty)\), Range: \([2, +\infty)\)